I cut a half inch wide strip of Gorilla Tape to secure the cut end, extending halfway around the bars. Then a full wrap 3M electrical tape, with a very slight stretch so the sides of the 3M tape look good. That's easy and effective for me. Gorilla tape doesn't creep like regular electrical tape. And I use a small piece of Gorilla tape to secure the starting end at the handlebar plug area, then I can start pulling on the bar tape tension right away -- easy.
(looking at the bar tops with a flashlight now, I can make out the texture of the gorilla tape through the 3M wrap. But I never notice that in "real life".)
It's the thicker, better 3M "Super 33+" tape. And the mini sized "Gorilla Tape To Go", 1 inch by 10 yards. I cut a strip in half lengthwise for this.